The Braves were in line to win the game in the ninth but Josh Walden surrendered a 5-4 lead when Carlos Gonzalez (5-for-5, two RBI) delivered a one-out RBI single, forcing an extra inning.
Atlanta responded in the 10th when Uggla drew a lead-off walk from Edgmer Escalona (1-4) and Simmons (3-for-5, two RBI) followed by lining a shot into the left-center field gap, easily scoring the game-winning run.
Uggla and Chris Johnson each drove in two runs for the National League East-leading Braves, who racked up a fourth straight win in the first of a four-game series.
Scott Downs (1-0), obtained in a trade from the Los Angeles Angels earlier Monday, fired 1 1/2 innings of scoreless relief to earn the victory.
Wilin Rosario went 3-for-5 with a home run while Nelson Arenado also had a home run for Colorado, which had 15 hits in the loss.